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Spatial Interpolation Methods Page | 0 of 0 SPATIAL INTERPOLATION METHODS 2018 Page | 1 of 1 1. Introduction Spatial interpolation is the procedure to predict the value of attributes at unobserved points within a study region using existing observations (Waters, 1989). Lam (1983) definition of spatial interpolation is “given a set of spatial data either in the form of discrete points or for subareas, find the function that will best represent the whole surface and that will predict values at points or for other subareas”. Predicting the values of a variable at points outside the region covered by existing observations is called extrapolation (Burrough and McDonnell, 1998). All spatial interpolation methods can be used to generate an extrapolation (Li and Heap 2008). Spatial Interpolation is the process of using points with known values to estimate values at other points. Through Spatial Interpolation, We can estimate the precipitation value at a location with no recorded data by using known precipitation readings at nearby weather stations. Rationale behind spatial interpolation is the observation that points close together in space are more likely to have similar values than points far apart (Tobler’s Law of Geography). Spatial Interpolation covers a variety of method including trend surface models, thiessen polygons, kernel density estimation, inverse distance weighted, splines, and Kriging. Spatial Interpolation requires two basic inputs: · Sample Points · Spatial Interpolation Method Sample Points Sample Points are points with known values. Sample points provide the data necessary for the development of interpolator for spatial interpolation. The number and distribution of sample points can greatly influence the accuracy of spatial interpolation. 2. Spatial Interpolation Methods Point interpolation methods may be categorized in several ways. · local or global · exact or approximate Reference # Page | 2 of 2 · gradual or abrupt · deterministic or stochastic Global Interpolation uses every known point available to estimate an unknown value. Local Interpolation uses a sample of known points to estimate an unknown value. This method is designed to capture the local or short range variation. Exact interpolation predicts a value at the point location that is the same as is known value. Approximate interpolation (inexact) predicts a value at the point location that differs from its known value. Deterministic interpolation method provides no assessment of errors with predicted values. Stochastic interpolation methods offer assessment of prediction errors with estimated variance. Gradual or abrupt interpolators are distinguished by the continuity of the surface they produce. Gradual interpolation techniques will produce a smooth surface with gradual changes occurring between observed data points. A wide variety of spatial interpolation methods exist in the literature. The spatial interpolation methods covered in this review are only those commonly used in the studies. 2.1.Theissen polygons Theissen polygons are an exact method of interpolation that assumes the unknown values of the points on a surface to be equal to the value of the nearest known point. This can be considered as a local interpolator because the characteristics of the global data set have no influence on the interpolation process. The steps to follow are: 1. Construct the Thiessen Polygons 2. Determine the area proportions of each polygon and assign a weight 3. Calculate the average value by using the weights (Thiessen, 1911) The method of Theissen polygons is a robust technique and will always produce the same surface from the same set of data points. This is a disadvantage that the technique is unintelligent and unable to respond to external knowledge about factors which may influence the values recorded at the observed data points. One of the most common uses for this technique is for the generation of area territories from a set of points. Reference # Page | 3 of 3 2.2.The Triangulated Irregular Network (TIN) The Triangulated Irregular Network (TIN) developed in the early 1970’s is a simple way to construct a surface from a set of known points. It is a particularly useful technique for irregularly spaced points. The TIN approach is an exact interpolation method. In the TIN model, the known data points are connected by lines to form a series of triangles. A TIN is a terrain model type that creates a set of continuous, non-overlapping, connected triangles (faces), based on a so-called Delaunay triangulation of irregularly spaced observations. The corners of the triangles are identical to the observations and within each triangle the surface is usually represented by a plane. A TIN model will allow for different density in the data model in different areas. The size of the triangles can therefore be adjusted to reflect the degree of relief in the surface to be modelled, provided more data has been gathered in areas of variable elevation characteristics. The use of triangles ensures that each piece of the surface fits the neighbouring pieces, and it is thus possible to create the desired continuous seamless transition between the triangles, within the model. 2.3. Trend Surface Models An approximate method, trend surface analysis approximates points with known values with a polynomial equation. A surface interpolation method that fits a polynomial surface by least-squares regression through the sample data points. This method results in a surface that minimizes the variance of the surface in relation to the input values. The resulting surface rarely goes through the sample data points. This is the simplest method for describing large variations, but the trend surface is susceptible to outliers in the data. Trend surface analysis is used to find general tendencies of the sample data, rather than to model a surface precisely (ESRI). The linear equation or the interpolator can then be used to estimate values at other points. 푍(푥, 푦) = 푏0+푏1푥 + 푏2푦 Where the attribute value z is a function of x and y coordinates. The b coefficients are estimated from the known points. The quadratic interpolator can be written as given below. 2 2 푍(푥, 푦) = 푏0+푏1푥 + 푏2푦 + 푏3푥 + 푏4푦 + 푏5푥푦 Reference # Page | 4 of 4 The two general classes of techniques for estimating a regular grid of points on a surface from scattered observations are methods called "global fit" and "local fit." As the name suggests, global- fit procedures calculate a single function describing a surface that covers the entire map area. The function is evaluated to obtain values at the grid nodes. In contrast, local-fit procedures estimate the surface at successive nodes in the grid using only a selection of the nearest data points. Trend surface analysis is the most widely used global surface-fitting procedure. The mapped data are approximated by a polynomial expansion of the geographic coordinates of the control points, and the coefficients of the polynomial function are found by the method of least squares, insuring that the sum of the squared deviations from the trend surface is a minimum. Each original observation is considered to be the sum of a deterministic polynomial function of the geographic coordinates plus a random error. The polynomial can be expanded to any desired degree, although there are computational limits because of rounding error. The unknown coefficients are found by solving a set of simultaneous linear equations which include the sums of powers and cross products of the X, Y, and Z values. Once the coefficients have been estimated, the polynomial function can be evaluated at any point within the map area. It is a simple matter to create a grid matrix of values by substituting the coordinates of the grid nodes into the polynomial and calculating an estimate of the surface for each node. Because of the least-squares fitting procedure, no other polynomial equation of the same degree can provide a better approximation of the data. The trend surface method fits a surface to the known observation points by a method known as least squares. The shape of the surface is determined by the mathematical equation, or polynomial, which is used to describe it. A polynomial surface has some very important characteristics; most importantly it changes smoothly and predictably from one map edge to the other. The polynomial equation may be linear, quadratic or even cubic. A linear equation would be used to describe a tilted plane. A quadratic equation would define a simple hill or hollow and a cubic equation would define a more complex surface. The trend surface method fits a surface to the known observation points by a method known as least squares. The shape of the surface is determined by the mathematical equation, or polynomial, which is used to describe it. A polynomial surface has some very important characteristics; most importantly it changes smoothly and predictably from one map edge to the other. The polynomial equation may be linear, quadratic or even cubic. A linear equation would be used to describe a tilted plane. A quadratic equation would define a simple hill or hollow and a cubic equation would define a more complex surface. Reference # Page | 5 of 5 2.4. Nearest neighbor interpolation Nearest-neighbor interpolation is a simple method of multivariate interpolation in one or more dimensions. Interpolation is the problem of approximating the value of a function for a non-given point in some space when given the value of that function in points around (neighboring) that point. The nearest neighbor algorithm selects the value of the nearest point and does not consider the values of neighboring points at all, yielding a piecewise-constant interpolant. The algorithm is very simple to implement and is commonly used (usually along with mipmapping) in real-time 3D rendering to select color values for a textured surface. For a given set of points in space, a Voronoi diagram is a decomposition of space into cells, one for each given point, so that anywhere in space, the closest given point is inside the cell.
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